Adding an LNA in this situation isn't going to help much at all, and may make things worse.

An LNA helps if you have long coax runs (overcomes the coax loss), and at higher UHF frequencies where the noise figure becomes more important.

You really need to work on the antenna positioning first.

For the 120 - 160 Mhz band, the BCFM filter could help quite a bit. But it does depend on if you are receiving strong broadcast FM in your area. Tune to the BCFM band and see how strong the signals are. If they are strong (and they are for most locations since the TX power of broadcast FM is quite strong), then the BCFM filter will probably help.

The disadvantage of having a very broadband antenna with preamplifier is that without filter/frequency selective antenna the receiver and or the preamplifier in the antenna may be overloaded. This means the active components cannot process all signals with the same factor, e.g. weak and stong signals by 10 dB. This is called Intermodulation or IM, which will produce additional signals that are not present at the antenna (e.g. IM 3rd order 2*f1 +f2 and 2*f2 -f1 in case you have only IM of 2 strong sigals). You can identify this if you add a 10 or 20 dB attenuator. If you have IM than all signals that were at least a bit stronger than 10 or 20 dB in signals strength are gone you have IM.